| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cats1cld.1 |
|- T = ( S ++ <" X "> ) |
| 2 |
|
cats1cli.2 |
|- S e. Word _V |
| 3 |
|
cats1fvn.3 |
|- ( # ` S ) = M |
| 4 |
|
cats1fv.4 |
|- ( Y e. V -> ( S ` N ) = Y ) |
| 5 |
|
cats1fv.5 |
|- N e. NN0 |
| 6 |
|
cats1fv.6 |
|- N < M |
| 7 |
1
|
fveq1i |
|- ( T ` N ) = ( ( S ++ <" X "> ) ` N ) |
| 8 |
|
s1cli |
|- <" X "> e. Word _V |
| 9 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
| 10 |
5 9
|
eleqtri |
|- N e. ( ZZ>= ` 0 ) |
| 11 |
|
lencl |
|- ( S e. Word _V -> ( # ` S ) e. NN0 ) |
| 12 |
|
nn0z |
|- ( ( # ` S ) e. NN0 -> ( # ` S ) e. ZZ ) |
| 13 |
2 11 12
|
mp2b |
|- ( # ` S ) e. ZZ |
| 14 |
6 3
|
breqtrri |
|- N < ( # ` S ) |
| 15 |
|
elfzo2 |
|- ( N e. ( 0 ..^ ( # ` S ) ) <-> ( N e. ( ZZ>= ` 0 ) /\ ( # ` S ) e. ZZ /\ N < ( # ` S ) ) ) |
| 16 |
10 13 14 15
|
mpbir3an |
|- N e. ( 0 ..^ ( # ` S ) ) |
| 17 |
|
ccatval1 |
|- ( ( S e. Word _V /\ <" X "> e. Word _V /\ N e. ( 0 ..^ ( # ` S ) ) ) -> ( ( S ++ <" X "> ) ` N ) = ( S ` N ) ) |
| 18 |
2 8 16 17
|
mp3an |
|- ( ( S ++ <" X "> ) ` N ) = ( S ` N ) |
| 19 |
7 18
|
eqtri |
|- ( T ` N ) = ( S ` N ) |
| 20 |
19 4
|
eqtrid |
|- ( Y e. V -> ( T ` N ) = Y ) |